It depends upon what you define as a "Greek island". Santorini is famous for its spectacular views, but it is a small island and one of the most touristy and expensice islands in Greece. You can get an idea here:
If you mean an authentic and more laid back island there are many choices. Here is a blog from donnyb from Naxos to make you see the difference:
I am unclear on how many days you actually have. Assuming 7 nights, Athens and one island destination works. Santorini is certainly an amazing place- especially the caldera side. It is among the most beautiful places on earth I think. It might not be exactly an authentic Greek Island experience but it is a memorable and worthwhile one. You could easily fly to Chania Crete from Athens for four nights and explore that area. That experience would fit with your authentic Greek Island criteria.
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